I think you've set it out in your op - you need to keep up the regular contact, even if you feel like a bit of a creepy stalker.
I have found that the best conditions for making friends is when you have some reason to see the person regularly without making a special effort and there is an opportunity to relax and socialise. That's why making friends at school or uni works really well, why it's harder at work unless there's a very social culture at your workplace and even harder if you don't have work or study going on.
Secondly friendship groups are better at self sustaining than one on one relationships. So you can kind of manipulate these two factors to make new friends.
The issue is that a lot of adults are not actively seeking and pursuing friendships. Either because they already have a group of friends through work, school, uni etc or because they don't know that this is something you can consciously do.
So step one is to become a creature of habit. Find several regular events to go to where you'll meet people - a club, a weekly/fortnightly/monthly special event, a baby/toddler group if applicable, if nothing else just go to the same place at the same time every week. You'll start noticing when other people are there at the same time. That gives you a reason to chat a little more each time and get to know people.
Next once you've made a connection with a person or preferably several people, you can either add your own extra regular event or just keep inviting them to stuff outside of that regular meeting. Don't worry about seeming over keen, if they are accepting your invitations. If they keep making excuses maybe leave the ball in their court.
Thirdly make them feel part of a group. Identify two or more people you've met in a similar context or have things in common with e.g. Similar age DC, or met at the same place. Then, whenever you invite one person to something, include the other(s) as well. It doesn't matter if they can't always make it. But soon you'll find that they start to think of you all as a group so when one of them decides to invite the other for coffee, they automatically think to invite you as well. This multiplies your chance that somebody will be the kind of person who initiates activities.